The corresponding nonlinear optimization theory was developed
in e.g. [1]}, [2]}, [3]} as follows.
If the variation \(\partial _u(\partial ^\varepsilon _u \mathcal {E})\in {L}(V(\Omega _t), V(\Omega _t)^\star )\)  with respect to \(u\)  in (REF ) exists,
and the associated adjoint operator \([\partial _u(\partial ^\varepsilon _u \mathcal {E})(u^\varepsilon _t; \Omega _t)]^\star \in {L}(V(\Omega _t), V(\Omega _t)^\star )\) 
satisfying
\(\langle [\partial _u(\partial ^\varepsilon _u \mathcal {E}) (u^\varepsilon _t; \Omega _t)]^\star v, u \rangle =\langle \partial _u(\partial ^\varepsilon _u \mathcal {E}) (u^\varepsilon _t; \Omega _t)u, v \rangle \quad \text{for all } u,v\in V(\Omega _t)\) 
